## Description

1. Polynomial fitting. Suppose we observe pairs of points (ai

, bi), i = 1, . . . , m representing

measurements from a scientific experiment. The variables ai are the experimental

conditions and the bi correspond to the measured response in each condition. We fit a

degree p < m polynomial to these data. In other words, we want to find the coefficients

of a degree p polynomial w(a) so that w(ai) ≈ bi

for i = 1, 2, . . . , m.

a) Suppose w(a) is a degree p polynomial. Write the general expression for w(ai) =

bi

.

b) Express the i = 1, . . . , m equations as a system in matrix form Ax = d while

defining A and d. What is the form/structure of A in terms of the given ai?

c) Write a script to find the least-squares model fit to the m = 30 data points in

polydata.mat. Plot the points and the polynomial fits for p = 1, 2, 3.

2. Least Squares Approximation of Matrices.

a) Derive the solution to least-squares problem minw kx − T wk

2

2 when T is an nby-r matrix of orthonormal columns. Your solution should not involve a matrix

inverse.

b) Let X =

x1 x2 · · · xp

be an n-by-p matrix. Use the least-squares problems minwi

kxi − T wik

2

2

to find W =

w1 w2 · · · wp

in the approximation

X ≈ TW. Your solution should express W as a function of T and X.

3. We return to the movies rating problem of Activity 5. The ratings on a scale of 1-10

are:

Movie Jake Jennifer Jada Theo Ioan Bo Juanita

Star Trek 4 7 2 8 7 4 2

Pride and Prejudice 9 3 5 6 10 5 5

The Martian 4 8 3 7 6 4 1

Sense and Sensibility 9 2 6 5 9 5 4

Star Wars: Empire Strikes 4 9 2 8 7 4 1

A matrix X containing this data is available in the file movie.mat and the csv file

movie.csv. Our goal is to approximate X using r “tastes”, the columns of T , that

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is, X ≈ TW where T is 5-by-r. You will use a Gram-Schmidt orthogonalization

code to find a set of tastes that approximate the ratings. A script that implements

Gram-Schmidt orthogonalization is available.

Define a 5-by-r taste matrix T =

t1 t2 · · · tr

with orthonormal columns and

the r-by-7 weight matrix

W =

w11 w12 · · · w17

w21 w22 · · · w27

.

.

.

wr1 wr2 · · · wr7

a) In Activity 5 you found the baseline (average) rating for each friend by requiring

the first basis vector in the taste matrix to be

t1 =

1

√

5

1

1

1

1

1

You have noticed by now that the first vector in the Gram-Schmidt procedure

is a scaled version of the first vector of the matrix, so you decide to define an

augmented matrix X˜ =

1 X

where 1 is a column vector containing five

unity entries. Apply a Gram-Schmidt orthogonalization code to X˜ to find a set

of orthonormal basis vectors. Is the first basis vector you obtain equal to t1?

b) Use your solution to the preceding problem in this homework assignment to find

the rank-1 approximation of X using only t1. That is, find W so that X ≈ t1W.

Use W to compute t1W. This gives you each friend’s baseline ratings. Also

compute the residual error X − t1W.

c) Now find a rank-2 approximation using T =

t1 t2

. That is, find W so that

X ≈ TW. Use W to compute TW. This gives you a rank-2 approximation to

the ratings. Also compute the residual error X −TW. How does t2 relate to the

distinction between sci-fi and and romance movie preferences?

d) Now find a rank-3 approximation using T =

t1 t2 t3

. That is, find W so

that X ≈ TW. Use W to compute TW. This gives you a rank-3 approximation

to the ratings. Also compute the residual error X − TW. Qualitatively discuss

the effect of increasing the rank of the approximation on the residual error.

e) Suppose you interchange the order of Jake and Jennifer so that Jennifer’s ratings

are in the first column of X and Jake’s ratings are in the second column. Does the

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rank-2 approximation change? Why or why not? Does the rank-3 approximation

change? Why or why not?

4. Let Q =

”

1 0

0 2 #

.

a) Is Q 0?

b) Sketch the surface y = x

TQx where x =

”

x1

x2

#

. If you find 3-D sketching too

difficult, you may draw a contour map with labeled contours.

5. Suppose P 0 and Q 0 are (symmetric) positive definite n × n matrices. Prove

that QPQ 0.

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